These verbs fall into two classes - one is 'punctual', and the other 'change of state'

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Change of state - find a solution, cross the finish line - This involves a preliminary activity that culminates in the act denoted by the verb. Thus, a person searches before finding the solution, and runs toward the finish line before crossing it.

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With change of state verbs, progressive aspect may or may not be possible, depending on whether the activity leading up to the achievement is treated as being the same activity

For example,

'His train is arriving at noon.' is acceptable

'* She is recognizing the thief.' is not acceptable.

I feel that 'She is recognizing the thief.' is strange, but as I don't understand the bolded part, I can't get the reason. Would you help me with this?

Honestly, I think the author is just confused here. He starts by saying "Change of state ... This involves a preliminary activity that culminates in the act denoted by the verb.". Then goes on to use "recognize" as an example of change-of-state verb. What is the preliminary activity that culminates in "recognizing"? There isn't any. By the author's own definitions, "recognize" is a "punctual" verb.
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Ben LeeMay 14 '13 at 18:20

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And the real distinction is that progressive actions on what the author calls punctual verbs are non-nonsensical and/or awkward, whereas progressive actions on change-of-state verbs are natural. But it's important to note that these are the author's own definitions, and by no means restrictive and well-defined categories. "He is spotting the airplane" makes sense if he has had his eye to the sky waiting for an airplane to appear, but not if it was out of the blue. OTOH, since "recognize" carries with it a connotation of surprise, progressive aspect rarely makes sense for that verb.
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Ben LeeMay 14 '13 at 18:22

One final note: It's potentially misleading to call statements like that part of a "grammar". Nearly all verbs, regardless of meaning, can be expressed with a progressive aspect. Colorless green ideas are sleeping furiously..
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Ben LeeMay 14 '13 at 18:31

2 Answers
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"She is recognizing the thief" would not be acceptable because she is still in the process of, still trying to place the face with the situation or name. It infers that it is a slower cognitive and ongoing process.

EG: They own a cottage in Somerset. (The possession is a state and not
an action. We cannot write this sentence in the progressive aspect)

Thus, They are owning a cottage is considered by many to be wrong.

Linked but not directly to: He's spotting the airplane and She is recognizing the thief.

To spot = to see

To recognize = to see and understand

Usually the senses; hear, see, smell, taste are not normally used in the progressive tense.
For example, we don't say:

I am seeing a picture

instead we prefer

I can see a picture.

From wikipedia

Dynamic verbs have duration, that is, they occur over time. This time
may or may not have a defined endpoint, and may or may not yet have
occurred. These distinctions lead to various forms related to tense
and aspect. For example, a dynamic verb may be said to have a durative
aspect if there is not a defined endpoint, or a punctual aspect if
there is a defined endpoint.